On étudie les propriétés algébriques fondamentales d’un polynôme de Tutte à 3 variables que l’auteur a associé à un morphisme de matroïdes - plus précisément à un morphisme fort, ou perspective dans le présent article, ou encore, de façon équivalente d’après le Théorème de Factorisation, à un matroïde muni d’un sous-ensemble distingué d’éléments. La plupart des propriétés algébriques du polynôme de Tutte habituel à 2 variables se généralisent au polynôme à 3 variables. Parmi les propriétés spécifiques on montre que le polynôme à 3 variables d’un matroïde pointé par un sous-ensemble normal peut être utilisé pour raccourcir le calcul du polynôme de Tutte (à 2 variables) de , et que le polynôme de Tutte à 3 variables d’une perspective de matroïdes est équivalent pour le calcul aux polynômes de Tutte à 2 variables des matroïdes de sa factorisation de Higgs.
We study the basic algebraic properties of a 3-variable Tutte polynomial the author has associated with a morphism of matroids, more precisely with a matroid strong map, or matroid perspective in the present paper, or, equivalently by the Factorization Theorem, with a matroid together with a distinguished subset of elements. Most algebraic properties of the usual 2-variable Tutte polynomial of a matroid generalize to the 3-variable polynomial. Among specific properties we show that the 3-variable Tutte polynomial of a matroid pointed by a normal subset can be used to abridge the computation of the 2-variable Tutte polynomial of , and that the 3-variable Tutte polynomial of a matroid perspective is computationally equivalent to the two-variable Tutte polynomials of the matroids of its Higgs factorization.
@article{AIF_1999__49_3_973_0, author = {Las Vergnas, Michel}, title = {The {Tutte} polynomial of a morphism of matroids {I.} {Set-pointed} matroids and matroid perspectives}, journal = {Annales de l'Institut Fourier}, pages = {973--1015}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {49}, number = {3}, year = {1999}, doi = {10.5802/aif.1702}, mrnumber = {2000f:05024}, zbl = {0917.05019}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.1702/} }
TY - JOUR AU - Las Vergnas, Michel TI - The Tutte polynomial of a morphism of matroids I. Set-pointed matroids and matroid perspectives JO - Annales de l'Institut Fourier PY - 1999 SP - 973 EP - 1015 VL - 49 IS - 3 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.1702/ DO - 10.5802/aif.1702 LA - en ID - AIF_1999__49_3_973_0 ER -
%0 Journal Article %A Las Vergnas, Michel %T The Tutte polynomial of a morphism of matroids I. Set-pointed matroids and matroid perspectives %J Annales de l'Institut Fourier %D 1999 %P 973-1015 %V 49 %N 3 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.1702/ %R 10.5802/aif.1702 %G en %F AIF_1999__49_3_973_0
Las Vergnas, Michel. The Tutte polynomial of a morphism of matroids I. Set-pointed matroids and matroid perspectives. Annales de l'Institut Fourier, Tome 49 (1999) no. 3, pp. 973-1015. doi : 10.5802/aif.1702. http://www.numdam.org/articles/10.5802/aif.1702/
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